Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $t \neq 0$. $r = \dfrac{t^2 + 3t - 28}{-3t^2 + 18t + 81} \times \dfrac{4t - 36}{t + 7} $
Answer: First factor out any common factors. $r = \dfrac{t^2 + 3t - 28}{-3(t^2 - 6t - 27)} \times \dfrac{4(t - 9)}{t + 7} $ Then factor the quadratic expressions. $r = \dfrac {(t + 7)(t - 4)} {-3(t - 9)(t + 3)} \times \dfrac {4(t - 9)} {t + 7} $ Then multiply the two numerators and multiply the two denominators. $r = \dfrac { (t + 7)(t - 4) \times 4(t - 9)} { -3(t - 9)(t + 3) \times (t + 7)} $ $r = \dfrac {4(t + 7)(t - 4)(t - 9)} {-3(t - 9)(t + 3)(t + 7)} $ Notice that $(t - 9)$ and $(t + 7)$ appear in both the numerator and denominator so we can cancel them. $r = \dfrac {4(t + 7)(t - 4)\cancel{(t - 9)}} {-3\cancel{(t - 9)}(t + 3)(t + 7)} $ We are dividing by $t - 9$ , so $t - 9 \neq 0$ Therefore, $t \neq 9$ $r = \dfrac {4\cancel{(t + 7)}(t - 4)\cancel{(t - 9)}} {-3\cancel{(t - 9)}(t + 3)\cancel{(t + 7)}} $ We are dividing by $t + 7$ , so $t + 7 \neq 0$ Therefore, $t \neq -7$ $r = \dfrac {4(t - 4)} {-3(t + 3)} $ $ r = \dfrac{-4(t - 4)}{3(t + 3)}; t \neq 9; t \neq -7 $